Polynomial Equations and Inequalities: Fresh Take

Solving Real-World Applications of Polynomial Equations

The Main Idea
Polynomial equations show up everywhere in the real world—from calculating volumes and areas to modeling business profits and projectile motion. The key is translating the word problem into a polynomial equation, solving it, and then interpreting your answer in context.Remember, not all mathematical solutions make sense in real-world contexts. A negative length or a time before the experiment started might solve your equation but won’t answer the actual question!
Example: Volume of a Cone
A mound of gravel is shaped like a cone with height equal to twice the radius. The volume formula is [latex]V = \frac{2}{3}\pi r^3[/latex]. If a customer purchases 100 cubic feet of gravel, what are the radius and height of the cone?

Pro Tip: Always check if your answer makes sense in the real-world context. Negative distances, impossible times, or values outside reasonable bounds are clues you need to reconsider your solution.

Solving Polynomial Inequalities

The Main Idea
Polynomial inequalities ask “when is this polynomial positive?” or “when is it negative?”The key insight: polynomials only change sign at their zeros (where they cross the x-axis). So we find the zeros, then test the intervals between them.

Question Help: Solving Polynomial Inequalities

  1. Solve the related equation (set the polynomial equal to zero) to find the zeros.
  2. Plot these zeros on a number line—they divide the line into intervals.
  3. Choose a test value in each interval.
  4. Evaluate the polynomial at each test value to determine if it’s positive or negative in that interval.
  5. Select the intervals that satisfy your inequality.
  6. Write your answer in interval notation.

Solving a polynomial inequality not in factored form – use greatest common factor.

You can view the transcript for “Ex: Solve a Polynomial Inequality – Factor Using GCF (Degree 3)” here (opens in new window).

Solving a polynomial inequality not in factored form – factor a trinomial

You can view the transcript for “Ex: Solve a Polynomial Inequality – Factor a Trinomial (Degree 4)” here (opens in new window).

Example: Solve [latex](x + 3)(x + 1)^2(x - 4) > 0[/latex]

Recall: Use parentheses [latex]( )[/latex] for [latex]<[/latex] or [latex]>[/latex], and brackets [latex][ ][/latex] for [latex]\leq[/latex] or [latex]\geq[/latex]. The union symbol [latex]\cup[/latex] combines separate intervals.
Example: Find the domain of [latex]v(t) = \sqrt{6 - 5t - t^2}[/latex]

Pro Tip: For inequalities involving square roots or fractions, the polynomial inequality tells you where the function is defined (the domain).

Finding the Inverse of Invertible Polynomial Functions

The Main Idea
An inverse function “undoes” what the original function does. If [latex]f(x) = 5[/latex] when [latex]x = 2[/latex], then [latex]f^{-1}(5) = 2[/latex]. The inputs and outputs swap places!However, only one-to-one functions have inverses that are also functions. Most polynomials aren’t one-to-one, but some simple ones (like odd-degree polynomials) are.Warning: [latex]f^{-1}(x)[/latex] does NOT mean [latex]\frac{1}{f(x)}[/latex]! The notation [latex]f^{-1}[/latex] specifically means “inverse function,” not reciprocal.

Question Help: Finding an Inverse Function

  1. Verify the function is one-to-one (passes the horizontal line test).
  2. Replace [latex]f(x)[/latex] with [latex]y[/latex].
  3. Swap [latex]x[/latex] and [latex]y[/latex].
  4. Solve for [latex]y[/latex].
  5. Replace [latex]y[/latex] with [latex]f^{-1}(x)[/latex].
Example: Find the inverse of [latex]f(x) = 5x^3 + 1[/latex]

Recall: The graphs of [latex]f[/latex] and [latex]f^{-1}[/latex] are reflections of each other across the line [latex]y = x[/latex]. If [latex](a, b)[/latex] is on [latex]f[/latex], then [latex](b, a)[/latex] is on [latex]f^{-1}[/latex].
Pro Tip: Check your work by composing the functions: [latex]f(f^{-1}(x)) = x[/latex] and [latex]f^{-1}(f(x)) = x[/latex] should both be true.

Restricting the Domain to Find Inverses

The Main Idea
Most polynomial functions aren’t one-to-one over their entire domain. A parabola, for example, fails the horizontal line test.But if we restrict the domain (only use part of the graph), we can create a one-to-one function that does have an inverse.Think of it like only walking up one side of a hill—that way, each height corresponds to exactly one location on your path.

Question Help: Restricting Domain to Find an Inverse

  1. Identify where the function is one-to-one (often one side of the vertex for quadratics).
  2. Restrict the domain to that interval.
  3. Replace [latex]f(x)[/latex] with [latex]y[/latex].
  4. Swap [latex]x[/latex] and [latex]y[/latex].
  5. Solve for [latex]y[/latex]—you may get [latex]\pm[/latex] from a square root.
  6. Choose the sign ([latex]+[/latex] or [latex]-[/latex]) that matches your restricted domain.
  7. Write the inverse with its domain restriction.
Example: Find the inverse of [latex]f(x) = (x - 4)^2[/latex] with domain [latex]x \geq 4[/latex]

Example: Restrict the domain and find the inverse of [latex]f(x) = (x - 2)^2 - 3[/latex]

Pro Tip: For quadratics in vertex form, the vertex tells you where to split the domain. Choose one side of the vertex to restrict to.
Recall: The domain of [latex]f[/latex] becomes the range of [latex]f^{-1}[/latex], and the range of [latex]f[/latex] becomes the domain of [latex]f^{-1}[/latex]. They swap!